Causal Inference for Observational Longitudinal Survival Data
Causal vs. conditional effects
Randomized treatment
Assume a randomized treatment \(A\). Then the subpopulation with \(A = a\) is representative of what would have happened to the rest of the population, had they been treated.
Scale up subpopulation to full population size with weights \[ W_i^a = \frac{1(A_i = a)}{P(A = a)} \]
An estimate of the probability of survival, had everyone received treatment \(a\): \[ P(T^a > t) = \frac{1}{n} \sum_{i=1}^n W_i^a 1(T_i > t) \]
The \(W_i^a\) are referred to as inverse probability of treatment weights (IPTW).
Confounding
Suppose now that a confounder \(X \in \{0, 1\}\) (low/high risk) affects both treatment and outcome (non-randomized data).
If \(X\) is the only confounder, then we still have representativity within levels of \(X\).
So we assign different weights depending on \(X\): \[ W_i^a = \frac{1(A_i = a)}{P(A = a \mid X_i)} \]
This works regardless of the dimension of \(X\), and also for continuous covariates.
Time-varying treatment
Consider a case where treatment is a binary counting process \(A(t)\) that starts at 0 and jumps to 1 at time \(\tau\) when a treatment is undertaken (e.g. surgery).
Covariates \(X(t)\) are also time-dependent.
Can generalize to a weight process \[ W_i(t) = \frac{1}{\lambda^A(\tau \mid \mathcal{F}_{\tau-})^{A(t)} \mathrm{e}^{-\Lambda^A(t \wedge \tau)}} \] where \(\lambda^A\) is the hazard/rate of treatment initiation and \[ \Lambda^A(t) = \int_0^t \lambda^A(s \mid \mathcal{F}_{s-}) \mathrm{d} s \]
The processes \(W_i(t)\) are quite nice (exponential martingales), and can be estimated from data.
A data example
Observational data (\(n = 601\)) from oncologists at Rigshospitalet.
- \(T\) time from diagnosis of incurable throat cancer to death.
- \(A(t)\) insertion of metalic tube (stent) into throat.
- A bunch of covariates \(X(t)\), mostly measured at baseline.
Weighting does not change the conclusion much. But we did not have enough information about time-dependent confounders.
